Band-Limited White Noise

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Description

Simulation of White Noise

The Band-Limited White
Noise block generates normally distributed random numbers that are
suitable for use in continuous or hybrid systems.

Theoretically, continuous white noise has a correlation time
of 0, a flat power spectral density (PSD), and a total energy of infinity.
In practice, physical systems are never disturbed by white noise,
although white noise is a useful theoretical approximation when the
noise disturbance has a correlation time that is very small relative
to the natural bandwidth of the system.

In Simulink® software, you can simulate the effect of white
noise by using a random sequence with a correlation time much smaller
than the shortest time constant of the system. The Band-Limited White
Noise block produces such a sequence. The correlation time of the
noise is the sample rate of the block. For accurate simulations, use
a correlation time much smaller than the fastest dynamics of the system.
You can get good results by specifying

tc≈11002πfmax,

where fmax is the
bandwidth of the system in rad/sec.

Comparison with the Random Number Block

The primary
difference between this block and the Random Number block is that
the Band-Limited White Noise block produces output at a specific sample
rate. This rate is related to the correlation time of the noise.

Usage with the Averaging Power Spectral Density Block

The Band-Limited White Noise block specifies a two-sided spectrum,
where the units are Hz. The Averaging Power Spectral Density block
specifies a one-sided spectrum, where the units are the square of
the magnitude per unit radial frequency: Mag^2/(rad/sec). When you
feed the output of a Band-Limited White Noise block into an Averaging
Power Spectral Density block, the average PSD value is π times
smaller than the Noise power of the Band-Limited
White Noise block. This difference is the result of converting the
units of one block to the units of the other: 1/(1/2)(2π) =
1/π, where:

1/2 is the factor for converting from a two-sided
to one-sided spectrum

2π is the factor for converting from Hz to rad/sec

Algorithm

To produce the correct intensity of this noise, the covariance
of the noise is scaled to reflect the implicit conversion from a continuous
PSD to a discrete noise covariance. The appropriate scale factor is 1/tc,
where tc is the correlation time of the noise. This scaling ensures
that the response of a continuous system to the approximate white
noise has the same covariance as the system would have to true white
noise. Because of this scaling, the covariance of the signal from
the Band-Limited White Noise block is not the same as the Noise
power (intensity) parameter. This parameter is actually
the height of the PSD of the white noise. This block approximates
the covariance of white noise as the Noise power divided
by tc.